= Solution
The relation $Rm=mR^{-1}$ implies $mR^j=R^{-j}m$. Move each reflection generator to the right in any word, cancel pairs using $m^2=I$, and reduce exponents modulo $N$. Every element then has a normal form $R^j$ or $R^jm$, $0\leq j<N$.
For the actual <symmetry group> of a regular polygon these are $N$ distinct rotations and $N$ distinct reflections. The determinant distinguishes rotations from reflections in the planar action. Hence the source's <dihedral group> $D_N$ has order $2N$, with its subscript counting polygon vertices. The calculations below use that convention.
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